Shimura correspondence

By a modular form of weight $k$ one understands a function $f$ on the upper half-plane satisfying $f(\gamma z)=\chi(\gamma)(cz+d)^kf(z)$ for some suitable function $\chi:\Gamma\to\textbf{C}^{\times}$ when

\begin{equation}\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\end{equation}

is an element of some congruence subgroup of $\text{SL}(2,\textbf{Z})$ (cf. also Modular function).

If $k$ is an integer, E. Hecke defined operators $T_n$ for every integer $n$, and showed they could be simultaneously diagonalizable (cf. also Hecke operator). The $L$-series of a simultaneous eigenfunction (cf. also Dirichlet $L$-function) is then an Euler product.

Modular forms of half-integral weight arise naturally, for example as theta-series. A theta-series in $r$ variables is a modular form of weight $r/2$.

If $k$ is a half-integer, $T_n$ can only be defined if $n$ is a square on forms of weight $k$, and there is not enough information in the Hecke eigenvalues to determine the Fourier coefficients. The coefficients are not multiplicative, so the $L$-series is not an Euler product.

Using the Rankin–Selberg method and a converse theorem, G. Shimura [a1] showed that if $\widetilde{f}$ is a modular form of weight $k+1/2$, then there is a corresponding modular form of weight $2k$ such that the $T_{n^2}$ Hecke eigenvalue on $\widetilde{f}$ agrees with the $T_n$ Hecke eigenvalue of $f$.

This result was complemented by the important theorem of J.-L. Waldspurger [a2], showing that the $D$th Fourier coefficient of $\widetilde{f}$ agrees with $L(k/2,f,\chi D)$. Waldspurger also gave interpretations of these special values as periods of $f$ (integrals over over geodesics). W. Kohnen and D. Zagier [a3] gave a particularly useful treatment of a special case. Also useful is [a4]. P. Sarnak and S. Katok [a5] found similar results for Maass forms.

Given Waldspurger's theorem, the case where $k=1$ becomes particularly interesting, since if $f$ is the modular form of weight two associated with an elliptic curve, $L(1,f,\chi D)$ has an interpretation in terms of the Birch–Swinnerton-Dyer conjecture. The period interpretation of the special values is then connected with the work of B.H. Gross, Kohnen and Zagier [a6] on heights of Heegner points. A beautiful application of this connection with the Birch–Swinnerton-Dyer conjecture to the classical problem of computing the set of areas of rational right triangles was given in [a7].

An interesting approach to the Shimura correspondence and Waldspurger's theorem is offered by the theory of Jacobi forms, in which both $\widetilde{f}$ and its correspondent $f$ may be related to automorphic forms on the Jacobi group. See [a8] and [a9]; cf. also Automorphic form.

A. Weil realized that (Siegel) modular forms, particularly theta-series, should be interpreted as automorphic forms not on $\text{Sp}(2n)$, but on a certain double cover $\widetilde{\text{Sp}}(2n)$, the so-called metaplectic group. If $n=1$, then $\text{Sp}(2n)=\text{SL}(2)$, and this is the proper framework for understanding the classical Shimura correspondence, which can be regarded as a lifting from either $\widetilde{\text{SL}}(2)$ to $\text{PGL}(2)=O(3)$, or from $\widetilde{\text{GL}}(2)$ to $\text{GL}(2)$.

T. Kubota and K. Matsumoto constructed metaplectic covers of more general groups, provided the ground field contains sufficiently many roots of unity. The Shimura correspondence in this context is a lifting from automorphic forms on the covering group to automorphic forms on $G$ or (sometimes) its dual, obtained by reversing the long and short roots and interchanging the fundamental group with the dual of the centre. See [a10], [a11], [a12], [a13], [a14] for the Shimura correspondence on higher covers of higher rank groups. Finding analogues of Waldspurger's theorem in this context is an important open problem (as of 2000).

References

[a1]	G. Shimura, "On modular forms of half integral weight" Ann. of Math. , 97 (1973) pp. 440–481
[a2]	J.-L. Waldspurger, "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" J. Math. Pures Appl. , 60 (1981) pp. 375–484
[a3]	W. Kohnen, D. Zagier, "Values of $L$-series of modular forms at the center of the critical strip" Invent. Math. , 64 (1981) pp. 175–198
[a4]	I. Piatetski–Shapiro, "Work of Waldspurger" , Lie Group Representations II , Lecture Notes in Mathematics , 1041 , Springer (1984)
[a5]	P. Sarnak, S. Katok, "Heegner points, cycles and Maass forms" Israel J. Math. , 84 (1993) pp. 193–227
[a6]	B.H. Gross, W. Kohnen, D. Zagier, "Heegner points and derivatives of $L$-series II" Math. Ann. , 278 (1987) pp. 497–562
[a7]	J.B. Tunnell, "A classical Diophantine problem and modular forms of weight $3/2$" Invent. Math. , 72 (1983) pp. 323–334
[a8]	M. Eichler, D. Zagier, "Jacobi forms" , Birkhäuser (1985)
[a9]	D. Ginzburg, S. Rallis, D. Soudry, "A new construction of the inverse Shimura correspondence" Internat. Math. Res. Notices , 7 (1997) pp. 349–357
[a10]	D. Kazhdan, S.J. Patterson, "Towards a generalized Shimura correspondence" Adv. Math. , 60 (1986) pp. 161–234
[a11]	Y.Z. Flicker, "Automorphic forms on covering groups of $\text{GL}(2)$" Invent. Math. , 57 : 2 (1980) pp. 119–182
[a12]	Y.Z. Flicker, D. Kazhdan, "Metaplectic correspondence" Publ. Math. IHES , 64 (1986) pp. 53–110
[a13]	D. Bump, J. Hoffstein, "On Shimura's correspondence" Duke Math. J. , 55 (1987) pp. 661–691
[a14]	D. Savin, "Local Shimura correspondence" Math. Ann. , 280 (1988) pp. 185–190